So far we have only examined the derivative of a function at a certain number . If we move from the constant to the variable , we can calculate the derivative of the function as a whole, and come up with an equation that represents the derivative of the function at any arbitrary value. For clarification, the derivative of at is a number, whereas the derivative of is a function.
Graph of cubic function illustrating use of associated quadratic. When there is exactly 1 value of that gives When there are 3 values of that give When there are exactly 2 values of that give Point is a stationary point.
In the diagram there is a stationary point at
When there are exactly 2 values of
that produce
Aim of this section is to derive the condition that produces exactly 2 values of
See Cubic function as
product
of linear function and quadratic.
The associated quadratic when
Divide by
This division gives a quotient of and remainder of
Factor divides exactly. Therefore, remainder
the desired condition.
Consider the value specifically
. L'Hôpital's rule cannot be used here because
is what we are trying to find.
taken times.
We will look at the expression for different values of
with (approx) in which case the expression becomes
where is or taken
times. This approach is used here because function sqrt() can be written so
that it does not depend on logarithmic or exponential operations.
accurate to 21 places of decimals, not bad for one line of simple python code using high-school math.
This method for calculation of supposes that function sqrt() is available.
Programming language python interprets expression a**b as
Therefore, in python, can be calculated in accordance with the basic definition above:
# python code>>>getcontext().prec101# Precision of 101.>>>dx=Decimal('1E-50')>>>(2**dx-1)/dxDecimal('0.69314718055994530941723212145817656807550013436026')# ln(2)>>>(2**(-dx)-1)/(-dx)Decimal('0.693147180559945309417232121458176568075500134360253')# ln(2)>>>
When
The base of natural logarithms is the value of that gives
This value of , usually called
>>># python code>>>N=e=Decimal(1).exp();NDecimal('2.71828182845904523536028747135266249775724709369995957496697')>>>([nfornin(N,)forpinrange(0,70)fornin(n.sqrt(),)][-1]-1)*(2**70)Decimal('1.0000_0000_0000_0000_0000_0423_51647362715016770651530003719651328')>>># ln(e) = 1. Our calculation of ln(e) is accurate to 21 places of decimals.
The first derivative of or
As shown above, at any point gives the slope of
at point
is the slope of when
Figure 1: Diagram illustrating relationship between and When both and slope of When both and slope of When both and slope of Point is absolute minimum.
In the example to the right, and
Of special interest is the point at which or slope of
When and
The point is called a critical point or stationary point of
Because has exactly one solution for has exactly one critical point.
The value of at point is less than at both
Therefore the critical point is a minimum of
In this curve the point is both local minimum and absolute minimum.
Figure 1: Diagram illustrating relationship between and When or both and slope of When both and slope of When both and slope of When both and slope of Point is local maximum. Point is local minimum.
In the example to the right, and
Of special interest are the points at which or slope of
When and or
When and
The points are critical or stationary points of
Because has exactly two real solutions for has exactly two critical points.
Slope of to the left of is positive and
adjacent slope of to the right of is negative.
Therefore point is local maximum. Point is not absolute maximum.
Adjacent slope of to the left of is negative and
slope of to the right of is positive.
Therefore point is local minimum. Point is not absolute minimum.
Figure 1(a): Graph of and for with axis compressed. For maximum and
A cylindrical water heater is standing on its base on a hard rubber pad that is a perfect thermal insulator.
The vertical curved surface and the top are exposed to the free air. The design of the cylinder requires that the volume of the cylinder
should be maximum for a given surface area exposed to the free air. What is the shape of the cylinder?
Let the height of the cylinder be and let where
is the radius and is a constant.
Figure 1(c): Plan of county road between Town A and Town B to be constructed so that cost is minimum.
Town B is 40 miles East and 50 miles North of Town A. The county is going to construct a road from Town A to Town B.
Adjacent to Town A the cost to build a road is $500k/mile.
Adjacent to Town B the cost to build a road is $200k/mile.
The dividing line runs East-West 30 miles North of Town A.
Calculate the position of point C so that the cost of the road from Town A to Town B is minimum.
Let point
Then distance from Town A to point
Distance from Town B to point
Figure 1(d): Graphs showing cost of county road and Curve showing is actually . Cost is minimum where
Figure 1(e): Sheet of cardboard to be cut and folded to make box of maximum possible volume. Cut on purple lines, fold on red lines. Design of box includes top.
A piece of cardboard of length and width
will be used to make a box with a top. Some waste will be cut out of the piece of cardboard
and the remaining cardboard will be folded to make a box so that the volume of the box is maximum.
What is the height of the box?
Figure 1(f): Curves associated with design of cardboard box. and is maximum when
Figure 1: Graph of ellipse showing semi-chord through center. When length of is maximum, length of major axis When length of is minimum, length of minor axis
An ellipse with center at origin has equation:
Given values calculate:
length of major axis
length of minor axis.
In Figure 1 is any line from origin to ellipse and
is angle between axis and
Aim of this section is to calculate so that length of is maximum,
in which case length of major axis =
Figure 3: Curves and values associated with car jack. When When When
At what rate is point moving upwards:
(a) when ?
(b) when ?
(c) when ?
We have to calculate when is given.
(equation of circle)
For convenience we'll use the negative value of the square root and say that
Relative to line
When inchesminute.
When inchesminute.
When and
inchesminute.
This example highlights the mechanical advantage of this simple but effective tool. When the top of the jack is low, it moves quickly.
As the jack takes more and more weight, the top of the jack moves more slowly.
Figure 5: Image of analog clock showing minute and hour hands at o'clock.
An old fashion analog clock with American style face (12 hours) keeps accurate time.
The length of the minute hand is inches and the length of the hour hand is inches.
At what rate is the tip of the minute hand approaching the tip of the hour hand at 3 o'clock?
Let be distance between the two tips.
The task is to calculate
where is
angle subtended by side at center of clock.
Calculating
Angular velocity of minute hand radians/hour.
Angular velocity of hour hand radians/hour.
angular velocity of minute hand relative to hour hand radians/hour.
A multi-lane highway is oriented East-West. A vehicle is moving in the inside lane from West to East.
A law-enforcement officer with a radar gun is in position 50 feet South of the center of the inside lane.
When the vehicle is 200 feet from the radar gun, it shows the vehicle's speed to be 60 mph.
What is the actual speed of the vehicle?
Figure 2: Diagram showing position of piston as a function of rotation of crankshaft in reciprocating engine. Piston moves up and down between and inches.
Code supplied to grapher (without white space) is:
(5)(cos(x))+(((25)((cos(x))^2)+144)^(0.5))
When expressed in this way, it's easy to convert the code to python code:
(5)*(cos(x))+(((25)*((cos(x))**2)+144)**(0.5))
Positions of interest:
Graph and diagram of piston at top dead center.
Graph and diagram of piston at bottom dead center.
Graph and diagram of piston half-way between TDC and BDC.
Graph and diagram of crank half-way between TDC and BDC.
Graph and diagram of connecting rod tangential to crank.
Figure 3: Diagram showing velocity of piston as a function of position of piston in reciprocating engine. Strictly speaking, velocity At constant RPM, is constant. Therefore is used to illustrate velocity.
Figure 4a: Diagram showing acceleration of piston in reciprocating engine. Negative acceleration has a greater absolute value than the positive, but it does not last as long.
Acceleration introduces the second derivative. While velocity was the first derivative of position with respect to time,
acceleration is the first derivative of velocity or the second derivative of position.
From velocity above
By implicit differentation:
Substitute for and as defined above, and you see the code input to grapher
at top of diagram to right.
Figure 4b: Diagram showing "irregularities" in the curve of velocity while velocity is increasing. axis compressed to illustrate shape of curves.
"Kinks" in the curve:
It is not obvious by looking at the curve of velocity that there are slight irregularities in the curve when velocity is increasing.
However, the irregularities are obvious in the curve of acceleration.
During one revolution of the crankshaft there is less time allocated for negative acceleration than for positive acceleration.
Therefore, the maximum absolute value of negative acceleration is greater than the maximum value of positive acceleration.
Figure 5: Diagram showing positions of maximum velocity of piston in reciprocating engine. In the first quadrant, from to the piston moves through inches. In the second quadrant, from to the piston moves through inches. Therefore, in the first quadrant, acceleration must be greater than in the second quadrant.
Velocity is rate of change of position. See also Figure 3 above.
Minimum velocity:
Velocity is zero when slope of curve of position is zero.
This occurs at top dead center and at bottom dead center, ie, when
and
Maximum velocity:
Intuition suggests that the position of maximum velocity might be the point at which the connecting rod is tangent
to the circle of the crankshaft. In other words:
or, that the position of maximum velocity might be the point at which the piston is half-way between top dead center and bottom dead center.
In other words:
However, velocity is maximum when acceleration is
which occurs when
Suppose that the engine is rotating at radians/second or approx. RPM.
Figure 6: Diagram showing positions of minimum and maximum acceleration of piston in reciprocating engine.
Acceleration is rate of change of velocity. See also Figures 4a and 4b above.
Minimum acceleration:
Acceleration is zero when slope of curve of velocity is zero. This occurs at maximum velocity or when
is approx.
Maximum acceleration:
Acceleration is maximum when slope of curve of velocity is maximum.
Maximum negative acceleration occurs when slope of curve of velocity is maximum negative. This happens at top dead center
when
Maximum positive acceleration occurs when slope of curve of velocity is maximum positive. This happens before and after bottom
dead center when is approx.
Let the engine continue to rotate at radians/second.
abs inches/second or approx. times
the acceleration due to terrestrial gravity.
This maximum value of acceleration is maximum negative when
According to Newtonian physics , force = mass*acceleration, and
, work = force*distance. In this engine energy expended in just accelerating piston to maximum velocity
is proportional to rpm.
Perhaps this helps to explain why a big marine diesel engine rotating at low RPM can achieve efficiency of